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We study femtosecond spin currents through MgO tunneling barriers in CoFeB(2 nm)|MgO($d$)|Pt(2 nm) stacks by terahertz emission spectroscopy. To obtain transport information independent of extrinsic experimental factors, we determine the complex-valued spin conductance $\tilde{G}_d (\omega)$ of the MgO layer (thickness d= 0-6 Å over a wide frequency range $(\omega/2\pi=$ 0.5-8 THz). In the time $(t)$ domain,$ G_d (t)$ has an instantaneous and delayed component that point to (i) spin transport through Pt pinholes in MgO, (ii) coherent spin tunneling and (iii) incoherent resonant spin tunneling mediated by defect states in MgO. A remarkable signature of (iii) is its relaxation time that grows monotonically with $d$ to as much as 270 fs at $d= 6$ Å, in full agreement with an analytical model. Our results indicate that terahertz spin conductance spectroscopy will yield new and relevant insights into ultrafast spin transport for a wide range of materials.